Existence and approximation of solutions for Fredholm equations of the first kind with applications to a linear moment problem

The Cimmino algorithm is an interative projection method for finding almost common points of measurable families of closed convex sets in a Hilbert space. When applied to Fredholm equations of the first kind, the Cimmino algorithm produces weak approximations of solutions provided that solutions exist. We show that for consistent Fredholm equations of the first kind whose data satisfy some spectral conditions, the sequences produced by the Cimmino algorithm converge not only weakly but also in norm. Using this fact, we obtain an existence criterion for solutions to a class of moment problems and show that if problems in this class have solutions, then the Cimmino algorithm generate norm approximations of such solutions.